Moduli spaces of vector bundles, with a view toward coherent sheaves
Clay Mathematics Institute, October 6-10, 2006
Vector bundles have been a key topic in algebraic geometry since the inception of sheaf theory in the 1950s. More recently, they have also assumed a central role in mathematical physics as objects whose moduli spaces model important equations. Coherent sheaves are the `next level', more difficult to analyze and richer in information about the base variety.
The goal of the workshop is to advance a set of open directions in the theory, bringing together top contributors in diverse areas: algebraic, arithmetic, differential and symplectic geometry, Brauer groups of function fields with finite constant fields, integrable systems, mathematical physics, representation theory. Among the problems to be considered: topology of moduli spaces (generators and relations for the cohomology ring), Brill-Noether Theory, Verlinde formulas, syzygies and canonical maps, projective normality, the slope conjecture. In a more forward thrust, derived categories, Mukai flops, stable pairs and coherent systems, twisted sheaves, vector bundles over algebraic surfaces (VBAS!).
One occasion for the workshop is the extended visit of CMI Senior Scholar Peter E. Newstead. Professor Newstead of the University of Liverpool, a Ph.D. student of British algebraic geometer John A. Todd at the University of Cambridge and of Sir Michael Atiyah at the University of Oxford, is responsible, among other advancements, for foundational work on moduli spaces in the 1960s, as well as being the founder of the European network "Vector Bundles over Algebraic Curves" (VBAC)
Professor Newstead is the CMI scholar attached to the special vector-bundle semester organized jointly at Boston and Tufts Universities by Emma Previato, Montserrat Teixidor-i-Bigas and Loring Tu. The program is articulated in a series of team-taught weekly meetings held alternately at BU and Tufts as topics courses in mathematics.